Design: 500 kVA single phase transformer with tubes

Standard method (A.K. Sawhney). Half of each winding is on each limb, so ampere-turns per limb are $I_{HV}T_{HV}/2$. Yoke: $H_y/D_y = 1$ with depth equal to the largest stamping width. Axial length for reactance = mean of 402 and 377.5 mm = 389.75 mm.

i) Dimensions of core, window and yoke

Voltage per turn, $E_t = K\sqrt{Q}$ ($Q$ in kVA):

$$\begin{aligned} E_t &= 0.8\sqrt{500} = 17.889\ \text{V}\\ \phi_m &= \frac{E_t}{4.44f} = \frac{17.889}{4.44\times50} = 0.08058\ \text{Wb}\\ A_i &= \frac{\phi_m}{B_m} = \frac{0.08058}{1.5} = 0.05372\ \text{m}^2 = 537.2\ \text{cm}^2 \end{aligned}$$

For a cruciform (two-stepped) core with stacking factor 0.9, $A_i = 0.56d^2$ (standard ratio, Sawhney):

$$\begin{aligned} d &= \sqrt{\frac{A_i}{0.56}} = \sqrt{\frac{0.05372}{0.56}} = 0.3097\ \text{m} = 309.7\ \text{mm}\\ a &= 0.85d = 263.3\ \text{mm}\\ b &= 0.53d = 164.2\ \text{mm} \end{aligned}$$

Gross core area $A_{gi} = A_i/0.9 = 0.05969$ m².

Output equation (single phase): $Q = 2.22fB_m\delta K_wA_wA_i\times10^{-3}$

$$\begin{aligned} A_w &= \frac{Q}{2.22fB_m\delta K_wA_i\times10^{-3}}\\ &= \frac{500}{2.22\times50\times1.5\times2.75\times10^6\times0.27\times0.05372\times10^{-3}}\\ &= 0.07529\ \text{m}^2\\ W_w &= \sqrt{\frac{A_w}{2.5}} = 173.5\ \text{mm},\quad H_w = 2.5W_w = 433.8\ \text{mm}\\ D &= d + W_w = 483.3\ \text{mm (distance between adjacent limb centres)} \end{aligned}$$

Check: HV outside diameter 418 mm < $D$ = 483 mm and coil height 402 mm < $H_w$ = 434 mm, so the windings fit. The LV inside diameter (310 mm) only just clears the core circle (309.7 mm), so the LV is wound directly on a thin insulating cylinder over the core.

Yoke

Yoke height equal to its depth ($H_y/D_y = 1$), with depth equal to the largest stamping width:

$$\begin{aligned} D_y &= H_y = a = 263.3\ \text{mm}\\ A_{gy} &= a^2 = 0.06931\ \text{m}^2,\quad A_y = 0.9A_{gy} = 0.06238\ \text{m}^2 \end{aligned}$$

ii) Overall dimensions of frame

$$\begin{aligned} H &= H_w + 2H_y = 433.8 + 2(263.3) = 960.4\ \text{mm}\\ W &= D + a = 483.3 + 263.3 = 746.5\ \text{mm}\\ \text{Depth} &= D_y = a = 263.3\ \text{mm} \end{aligned}$$

 +-----------------------+  ---
 |       top yoke        |   ^
 +-----+-----------+-----+   |
 |limb |  window   |limb |   H
 |     |  Ww x Hw  |     |   |
 +-----+-----------+-----+   |
 |      bottom yoke      |   v
 +-----------------------+  ---
 |<--------- W --------->|
    |<----- D ----->|

iii) Per unit regulation at 0.8 pf lagging

Currents: $I_{LV} = \frac{Q}{V_{LV}} = 1250.00$ A; $I_{HV} = 75.758$ A.

$$\begin{aligned} T_{LV} &= \frac{V_{LV}}{E_t} = \frac{400.0}{17.889} = 22.36 \approx 22\\ E_t\ (\text{revised}) &= \frac{400.0}{22} = 18.182\ \text{V}\\ T_{HV} &= T_{LV}\frac{V_{HV}}{V_{LV}} = 363.0 \approx 363\\ a_{LV} &= \frac{I_{LV}}{\delta} = 454.55\ \text{mm}^2,\quad a_{HV} = \frac{I_{HV}}{\delta_{HV}} = 27.548\ \text{mm}^2 \end{aligned}$$

Radial depths: $b_1 = 19$ mm (LV), $b_2 = 29$ mm (HV), duct $a = 6$ mm. $L_c$ = 389.75 mm.

Mean lengths of turn:

$$\begin{aligned} L_{mt,LV} &= \pi\frac{310.0 + 348.0}{2} = 1.0336\ \text{m}\\ L_{mt,HV} &= \pi\frac{360.0 + 418.0}{2} = 1.2221\ \text{m}\\ L_{mt} &= \pi\frac{310.0 + 418.0}{2} = 1.1435\ \text{m (both windings, for reactance)} \end{aligned}$$

Resistance referred to HV, with $\rho = 0.021\ \Omega\text{-mm}^2/\text{m}$ (copper at about 75 °C):

$$\begin{aligned} r_{HV} &= \frac{\rho T_{HV}L_{mt,HV}}{a_{HV}} = \frac{0.021\times363\times1.2221}{27.548} = 0.33817\ \Omega\\ r_{LV} &= \frac{0.021\times22\times1.0336}{454.55} = 0.0010505\ \Omega\\ R_p &= r_{HV} + r_{LV}\left(\frac{T_{HV}}{T_{LV}}\right)^2 = 0.33817 + 0.0010505\left(\frac{363}{22}\right)^2 = 0.62418\ \Omega\\ \varepsilon_r &= \frac{I_{HV}R_p}{V_{HV}} = \frac{75.76\times0.62418}{6600.0} = 0.00716\ \text{p.u.} \end{aligned}$$

Leakage reactance (per unit), with $AT$ per limb $= 13750$ and axial winding length $L_c = 389.8$ mm:

$$\begin{aligned} \varepsilon_x &= 2\pi f\mu_0\frac{AT}{L_c}\cdot\frac{L_{mt}}{E_t}\left(a + \frac{b_1+b_2}{3}\right)\\ &= 2\pi(50)(4\pi\times10^{-7})\frac{13750}{0.3897}\cdot\frac{1.1435}{18.182}\left(6 + \frac{19+29}{3}\right)\times10^{-3}\\ &= 0.0193\ \text{p.u.} \end{aligned}$$ $$\begin{aligned} \varepsilon &= \varepsilon_r\cos\phi + \varepsilon_x\sin\phi\\ &= 0.00716(0.8) + 0.0193(0.600) = 0.0173\ \text{p.u.} = 1.73\% \end{aligned}$$

iv) Temperature rise and cooling tubes

Total loss $P = 1460 + 3865 = 5325$ W. Tubes assumed 50 mm in diameter and 1.5 m long (tank height 1.6 m).

Tank walls (top and bottom neglected), dissipation 12.5 W/m²°C (6 radiation + 6.5 convection):

$$\begin{aligned} S_t &= 2(W_t + L_t)H_t = 5.3440\ \text{m}^2\\ \theta_{plain} &= \frac{5325}{12.5S_t} = 79.72\ ^\circ\text{C} \end{aligned}$$

This exceeds 35 °C, so tubes are needed. Tubes improve convection by 35%: $6.5\times1.35 \approx 8.8$ W/m²°C.

$$\begin{aligned} A_{tubes} &= \frac{1}{8.8}\left(\frac{P}{\theta} - 12.5S_t\right) = \frac{1}{8.8}\left(\frac{5325}{35} - 12.5\times5.3440\right) = 9.698\ \text{m}^2\\ \text{Area of one tube} &= \pi d_tl_t = \pi\times0.05\times1.5 = 0.2356\ \text{m}^2\\ n_t &= \frac{9.698}{0.2356} = 41.16 \approx 42\ \text{tubes} \end{aligned}$$

Arrangement of tubes: with 50 mm tubes at 75 mm centre-to-centre spacing, one row round the tank can hold about $1050/75 = 14$ tubes on each long side and $620/75 \approx 8$ on each short side. So 42 tubes are placed in a single row: 14 on each long side and 7 on each short side ($2\times14 + 2\times7 = 42$).

        o o o o o o o o o o o o o o    (14)
      +-------------------------------+
   o  |                               |  o
   o  |                               |  o
   o  |        TANK (top view)        |  o
(7)o  |      1.05 m  x  0.62 m        |  o (7)
   o  |                               |  o
   o  |                               |  o
   o  |                               |  o
      +-------------------------------+
        o o o o o o o o o o o o o o    (14)
   o = cooling tube, 50 mm dia, 75 mm pitch

Answer: $d$ = 310 mm; window 174 × 434 mm; yoke 263 × 263 mm; frame 960 × 747 × 263 mm; regulation ≈ 1.73%; plain tank rise ≈ 79.7 °C; 42 tubes needed.